Rings of Frobenius operators

Abstract

Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component, and show that F(E) need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings

The Frobenius complexity of Hibi rings

“…To compute Frobenius complexity, we will be using the following correspondence [11], [15]. We state a version similar to Theorem 3.3 in [9], but note that we can state it more generally for C(R) which is isomorphic to F (E) op when R is complete and local. Theorem 2.4.…”

Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

“…In §7, we recall the operation T-construction defined by Katzman et al [KSSZ] and define the T-complexity of a commutative N-graded ring of characteristic p. By the result of Katzman et al [KSSZ,Theorem 3.3] and the results of previous sections, we see that the Frobenius complexity of a Hibi ring can be computed by the T-complexities of Ehrhart rings appeared in §5. We state key lemmas to compute the limit T-complexity of Ehrhart rings.…”

“…Definition 7.3 ([KSSZ, Definition 3.2]) Let R be a commutative Noetherian normal ring that is either complete local or N-graded and finitely generated over a field R 0 . Let ω denote the canonical ideal of R and for m ∈ Z, let ω (m) be the m-th power of ω in Div(R).…”

“….4 ([KSSZ, Theorem 3.3]) Let (R, m) be a commutative Cohen-Macaulay normal complete local ring of characteristic p, E the injective hull of R/m, R the anticanonical cover of R. Then there is an isomorphism of graded rings F (E) ∼ = T (R).Note in the setting of Fact 7.4, T (R) 0 = R and T (R) is a strong left R-skew algebra.…”

Discreteness of 𝐹-jumping numbers at isolated non-ℚ-Gorenstein points

“…[ELSV04, BdFFU15, Har06, BMS08, BMS09, KLZ09, BSTZ10, KZ14, ST14,KSSZ14]. In characteristic zero, discreteness and rationality of jumping numbers is elementary if X is Q-Gorenstein, but rationality fails in general [Urb12, Theorem 3.6].…”